We study second order sufficient optimality conditions (SSC) for optimal control problems with control appearing linearly. Specifically, time-optimal bang-bang controls will be in-], SSC have been developed in terms of the positive definiteness of a quadratic form on a critical cone or subspace. No systematical numerical methods for verifying SSC are to be found in these papers. In the present paper, we study explicit representations of the critical subspace. This leads to an easily

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... e test for SSC in the case of a bang-bang control with one or two switching points. In general, we show that the quadratic form can be simplified by a transformation that uses a solution to a linear matrix differential equation. Particular conditions even allow us to convert the quadratic form to perfect squares. Three numerical examples demonstrate the numerical viability of the proposed tests for SSC. Introduction. Second order sufficient optimality conditions (SSC) for optimal control problems subject to mixed control-state constraints have been studied by various authors; cf. Dunn [8, 9] ; Malanowski [22]; Maurer and Pickenhain [30]; Maurer and Oberle [29]; Milyutin and Osmolovskii [31]; Osmolovskii [35, 36, 37, 38, 39, 40]; and Zeidan [48] . SSC amount to testing the positive definiteness of a certain quadratic form on the so-called critical cone or subspace. Provided that the strict Legendre-Clebsch condition holds, a well-known numerical recipe allows the conversion of the quadratic form to a perfect square. Namely, it suffices to check that an associated Riccati matrix differential equation has a bounded solution along the extremal trajectory. This test has been performed in a number of practical examples and has played a crucial role in sensitivity analysis of parametric control problems; cf., e.g., Augustin, Malanowski, and Maurer [2, 21, 22, 23, 24, 25, 27, 28] . Recently, the Riccati approach has been also extended to discontinuous controls (broken extremals) by Osmolovskii and Lempio [42] . The above mentioned tests for SSC are not applicable to optimal control problems with control appearing linearly. Bang-bang controls do belong to this class of problems. Though first and higher order necessary optimality conditions for bangbang controls have been studied, e.g., in Bressan [3], Schättler [44] , and Sussmann *